1. **Problem statement:** Find the compositions $[f \circ g](x)$ and $[g \circ f](x)$ for the functions: $f(x) = x + 5$ and $g(x) = x - 3$ 2. **Recall the definition of composition:** $[f \circ g](x) = f(g(x))$ means substitute $g(x)$ into $f$. $[g \circ f](x) = g(f(x))$ means substitute $f(x)$ into $g$. 3. **Calculate $[f \circ g](x)$:** $$[f \circ g](x) = f(g(x)) = f(x - 3)$$ Substitute $x - 3$ into $f(x) = x + 5$: $$f(x - 3) = (x - 3) + 5$$ Simplify: $$= x - 3 + 5 = x + 2$$ 4. **Calculate $[g \circ f](x)$:** $$[g \circ f](x) = g(f(x)) = g(x + 5)$$ Substitute $x + 5$ into $g(x) = x - 3$: $$g(x + 5) = (x + 5) - 3$$ Simplify: $$= x + 5 - 3 = x + 2$$ 5. **Final answers:** $$[f \circ g](x) = x + 2$$ $$[g \circ f](x) = x + 2$$ Both compositions yield the same function $x + 2$.